Finding the limit of a function in f(x)-f(a)/x-a format

[noonan]

Homework Statement

Let f(x)=lim (csct-cscx)/t-x. Find the value of f'(pi/4)
t-x

Homework Equations

f(x)-f(a)/x-a

The Attempt at a Solution

I tried doing it from first principles but couldn't figure out how to get rid of h. I also tried doing L'hopital's rule and got root2 but I know the answer is 3root2. I also tried making the equation (cscx-root2)/x-(pi/4). Nothing seems to work!

 

[Mark44]

Homework Statement

Let f(x)=lim (csct-cscx)/t-x. Find the value of f'(pi/4)
t-x

Use parentheses around the terms in the denominator.

Homework Equations

f(x)-f(a)/x-a

Use parentheses around the numerator and denominator terms.

The Attempt at a Solution

I tried doing it from first principles but couldn't figure out how to get rid of h.

There is no h anywhere in your work.

I also tried doing L'hopital's rule and got root2 but I know the answer is 3root2. I also tried making the equation (cscx-root2)/x-(pi/4). Nothing seems to work!

Assuming that f(x) = csc(x), then f'($\pi/4$) is given by this limit.
$$\lim_{x \to \pi/4}\frac{csc(x) - csc(\pi/4)}{x - \pi/4}$$

My first step was to rewrite the csc terms using csc(x) = 1/sin(x). After that, I did some algebra to write the whole limit expression with a single numerator and a single denominator. If you are allowed to use L'Hopital's Rule, you get the answer pretty quickly.